The document discusses different types of t-tests, including dependent samples t-tests. It provides steps for hypothesis testing and examples of computing a dependent samples t-test. An example is shown comparing pretest and posttest scores from a study on enhancing students' self-concept through a special program. The t-value calculated supported rejecting the null hypothesis that the posttest mean was equal to the pretest mean, confirming the alternative hypothesis that the posttest mean was higher. The conclusion was that the self-concept program was effective in increasing students' self-concept.

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Compute by hand and interpret
Dependent samples t test
Single sample t test
Use SPSS to compute the same tests
and interpret the output

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The Slides discuss
• Comparing two means to ascertain which mean is of greater
statistical significance.
• The statistical test used for this type of analysis is the t test,
and the statistic that is computed is called a t value.
• Three research situations in which the t test can be used to
analyze the data and compare the means from
a) Two paired samples (e.g., pretest and posttest scores); and
b) a sample and a population (e.g., comparing a mean of a
sample to the mean of the population).
• Examples with numerical data to illustrate each of the three
types of t tests.
• After t value is computed, you have decided whether it is
statistically significant and whether your research
hypothesis was confirmed.
STEPS IN THE PROCESS OF HYPOTHESIS TESTING
1. Formulating a research question for the study.
2. Stating a research hypothesis (i.e., an alternative hypothesis). The hypothesis should represent the
researcher’s prediction about the outcome of the study and should be testable. Note that a null
hypothesis for the study is always implied, but it is not formally stated in most cases. The null
hypothesis predicts no difference between groups or means or no relationship between variables.
3. Designing a study to test the research hypothesis. The study’s methodology should include plans
for selecting one or more samples from the population that in interest to researcher; selecting or
designing instruments to gather the numerical data; carrying out the study’s procedure (and
intervention in experimental studies); and determining the statistical test(s) to be used to analyze
the data.
4. Conducting the study and collecting numerical data.
5. Analyzing the data and calculating the appropriate test statistics (e.g., Pearson correlation
coefficient, t test value or chi square value).
6. Determining the appropriate p value.
7. Deciding whether to retain or reject the null hypothesis. A p value of 0.05 is the most commonly
used benchmark to consider the results statistically significant. If the results are statistically
significant, the researcher may also wish to calculate the effect size (EF) to determine the
practical significance of the study’s results.
8. Making a decision whether to confirm the study’s alternative hypothesis (i.e., the research
hypothesis ) and how probable it is that the results were obtained purely by chance. This decision
is based on the decision made regarding the null hypothesis.
9. Summarizing the study’s conclusion, addressing the study’s research question.

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Assumptions of t-Test
• The groups are independent of each other.
• A person (or case) may appear in only one group.
• When the two group are approximately the same size,
there is no need for the homogeneity of variance. This
assumption is called the assumption of the homogeneity
of variance.
• We compare the two variances to determine if there is a
statistically significant difference between them.
• To test for the assumption of the homogeneity of the
variances, we divide the larger variance by the smaller
variance and obtain a ratio, called the F value.
• When the F value is statistically significant we cannot
assume that the variance are equal.
Assumptions of t-Test
• A test for the equality of variance, such as the levene’s is
used to test the significance of the F value. An F value
that is not statistically significant (p>0.05) indicates that
the assumption for the homogeneity of variance is not
violated and, therefore, equal variances can be assumed ,
on the other hand, a significant F value (p< 0.05)
indicates that the assumption for the homogeneity of
variance was violated. The t test statistical results are
adjusted for the unequal variance.

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Assumptions of t-Test
• The independent variables are interval or ratio.
• The population from which samples are drawn is
normally distributed.
• Samples are randomly selected.
• The groups have equal variance (Homogeneity of
variance).
• The t-statistic is robust (it is reasonably reliable even if
assumptions are not fully met). Therefore, even if the
assumption of the homogeneity of variance is not fully
met, the researcher can probably still use the test to
analyze the data.
A t-test is used to compare two means in three different situations
T Test
t-test for independent
samples
- The two groups whose
means are being
compared are
independent of each other
Ex: a comparison of
experimental and control
groups.
T-test for paired samples (t-test for
dependent, matched or correlation
samples)
The two means represent two
sets of scores that are paired.
Ex: a comparison of pretest
and posttest scores obtained
from one group of people
t-test for a single
sample
The t-test is used
when the mean of a
sample is compared
to the mean of a
population

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Dependent Samples
t-tests
Dependent Samples t-test
• A t test for paired samples (dependent sample t-test) is used
when the two means being compared come from two sets of
scores that are related to each other.
• It is used, for example, in experimental research to measure
the effect of an intervention (a treatment) by comparing the
posttest to the pretest scores.
• The most important requirement for conducting this t test is
that the two sets of scores are paired: they belong to the
same individuals.
• Useful to control individual differences. Can result in more
powerful test than independent samples t-test.

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Computing the paired-samples t-test
• First find for each person the difference (D) between the
two scores (e.g., between pretest and posttest) and sum up
those difference (∑ D). Usually, the lower scores (e.g.,
pretest) are subtracted from the higher scores (e.g., posttest)
so D values are always positive.
• We also need to compute the sum of the squares of the
difference (∑ D2). Finally the t value is computed using the
formula:
The example that follows demonstrates the computation of a paired-
samples t test. To simplify the computations , we use scores of eight
students only. Of course, in conducting real-life research, it is
recommended that larger samples (thirty or more) be used.

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An example of a t Test
for dependent samples or paired samples
Research has documented the potential effect of students’ positive self-concept on
their self-reflections, attitudes toward self, schoolwork, and general development. A
special program is developed by school psychologists and primary grade teachers to
enhance the self-concept of young-age children. The program is implemented in
Sunny Bay School with four groups of first-and second-grade students. The
intervention lasts six weeks and involves various activities in the class and at home.
The instrument used to assess the effectiveness of the program is comprised of forty
pictures, and scores can range from 0 to 40. The program coordinator conducts a
series of workshops to train five graduate students to administer the instrument. All of
the children in the program are tested before the start of the program and then again
during the first week after the end of the program. A t test for paired samples is used
to test the hypothesis that students’ self-concept would improve significantly on the
posttest, compared with their pretest scores. The research hypothesis is:
HA: Mean POSTTEST > Mean PRETEST
Ho: Mean POSTTEST = Mean PRETEST
Review 6 Steps for Significance
Testing
1. Set alpha (p level).
2. State the Research
question
3. Set up hypotheses,
Null and Alternative.
4. Calculate the test
statistic (sample
value).
4. Find the critical
value of the statistic.
5. State the decision
rule.
6. State the conclusion.

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Pretest and Posttest Scores of Eight Students
Pretest
X1
Posttest
X2
Posttes
t –
Pretest
D
(Posttest-Pretest)2
D2
30 31 +1 1
31 32 +1 1
34 35 +1 1
32 40 +8 64
32 32 0 0
30 31 +1 1
33 35 +2 4
34 37 +3 9
D= 17 D2 =81
 The table shows the numerical data we
used to compute the t value for the eight
students selected at random from the
program participants.
 The table shows the pretest and posttest
scores for each students, as well as the
means on the pretest and posttest.
 The third column in the table shows the
difference between each pair of scores
(D) and is created by subtracting the
pretest from the posttest for each
participant.
 The gain scores are then squared and
recorded in the fourth column (D2).
 The scores in these last two columns are
added up to create ∑D and ∑D2
respectively. These values are used in the
computation of the t value:
Dependent Samples t Example
1. Set alpha = .05
2. Null hypothesis: H0:  Posttest =
Pretest. Alternative is HA:  Mean
Posttest >  Pretest.
3. Calculate the test statistic:

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Dependent Samples t Example
4. Determine the critical value of t. Alpha =.05, one tailed
test df = N(pairs)-1 =8-1=7 ; Critical value is 1.895
5. Decision rule: The obtained t value of 2.37 exceeds the
critical values of 1.895 for one tailed test under p=0.05
6. Conclusion. We reject the null hypothesis that states that
there is no difference between the pretest and the posttest
scores. The chance that these results were obtained purely
by chance is less than 5 percent. We confirm the research
hypothesis that predicted that the posttest mean score
would be significantly higher than the pretest mean score.
Table A section from the Table of Critical
Value for t Tests
Level of significance (p value) for one-tailed test
p values 0.10 0.05 0.025 0.01 0.005
Level of significance (p value) for two-tailed test
p values 0.20 0.10 0.05 0.02 0.01
df=7 1.415 1.895 2.365 2.998 3.4991.895
0.05
df=7

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Interpretation
According to this study, the self-concept
enhancement program was effective in
increasing the self-concept of the first-and
second-grade students.
Our data seem to indicate that the intervention to increase the
self-concept of the primary grade students was effective.
However, those conducting the research or those reading
about it should still decide for themselves whether the
intervention is worth while. The question to be asked is
whether an increase of 2.13 (34.13-32.00) points (out of
forty possible points on the scale) is worth the investment of
time, money, and effort.
Using SPSS for dependent t-test
• Open SPSS
• Open file “SPSS Examples” (same as before)
• Go to:
– “Analyze” then “Compare Means”
– Choose “Paired samples t-test”
– Choose the two IV conditions you are
comparing. Put in “paired variables box.”

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Dependent t- SPSS output
Paired Samples Statistics
Mean N
Std.
Deviation
Std. Error
Mean
Pair 1 Pretest 32.00 8 1.604 .567
Postest 34.13 8 3.227 1.141
Paired Samples Correlations
N Correlation Sig.
Pair 1 Pretest & Postest 8 .635 .091
Paired Samples Test
Paired Differences
95% Confidence
Interval of the
Difference
Mean
Std.
Deviation
Std. Error
Mean Lower Upper
Pair
1
Pretest -
Postest
-2.125 2.532 .895 -4.242 -.008
Paired Samples Test
t df
Sig. (2-
tailed)
Pair
1
Pretest -
Postest
-2.374 7 .049
Task of t-test for dependent samples
The data below shows the lengths ( in arbitrary units)
of syllables containing a particular vowel in two
different environments. Each of the ten subjects was
asked to read two sentences, each containing the
vowel in one environment, and a pair of lengths was
thus obtained for each subject. The researcher predicts
greater mean length in environment 2 than in
environment 1.

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Subject no. Environment 1 Environment 2
1 22 26
2 18 22
3 26 27
4 17 15
5 19 24
6 23 27
7 15 17
8 16 20
9 19 17
10 25 30
Table of Lengths of a vowel in two
environments
Do 6 Steps for Significance Testing
1. Set alpha (p level).
2. State the Research
question
3. Set up hypotheses,
Null and Alternative.
4. Calculate the test
statistic (sample
value).
4. Find the critical
value of the statistic.
5. State the decision
rule.
6. State the conclusion.

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t-Tests for a Single
Sample
t-Test for a Single Sample
• A t test for single sample is used when the mean of a sample
is compared to the mean of a population.
• Occasionally, a researcher is interested in comparing a
single group (a sample) to a larger group (a population).
• For example, a high school teacher of a freshmen-
accelerated English class may want to confirm that the
students in that class had obtained higher scores on an
English placement test compared with their peers.
• In order to carry out this kind of a study the researcher must
know prior to the start of the study the mean value of the
population. In this example, the mean score of the
population is the overall mean of the scores of all freshmen
on the English placement test.

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Computing t-Test for Single sample
• The t value is computed using the formula:
An example of a t Test
for Single sample
A kindergarten teacher in a school commented to her colleague that the students in
her class this year seem to be less bright than those she had in the past. Her colleague
disagrees with her, and to test whether the first-graders this year are really different
from those in previous years, they conduct a test for a single sample. The scores used
are from the Wechsler Preschool and Primary Scale of Intelligence Third Edition
(WPPSI-III), which is given every year to all kindergarten students in the district. In
this example, we consider the district to be the population to which we compare the
mean of the current kindergarten class. Although the mean IQ score of the population
at large is 100 (µ=100), this district’s mean IQ score is 110 (µ= 110), and this mean is
used in the analysis. The research hypothesis is stated as a null hypothesis and
predicts that there is no statistically significant difference in the mean IQ score of this
year’s kindergarten students (the sample) and the mean IQ score of all kindergarten
students in the district (the population) that were gathered and recorded over the last
three years.
Ho: Mean Class = Mean District

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Review 6 Steps for Significance
Testing
1. Set alpha (p level).
2. State the Research
question
3. Set up hypotheses,
Null and Alternative.
4. Calculate the test
statistic (sample
value).
4. Find the critical
value of the statistic.
5. State the decision
rule.
6. State the conclusion.
Table IQ scores of Ten Students
Scores
115 118
135 113
105 98
107 120
112 99
∑X= 1,122
¯X=112.20
SD= 10.94
In order to test their
hypothesis, the two teachers
randomly select IQ scores of
ten students from this year’s
kindergarten class.
These IQ scores are listed in the
table followed by the
computation of the t value.

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Dependent Samples t Example
1. Is there any differences between this year’s
kindergarten class and the “typical kindergarten
students in the district?
2. Null hypothesis: H0:  Class =  District : there
is no significant difference between this year’s
kindergarten class and the “typical” kindergarten
students in the district.
3. Alternative is HA:  Class ≠  District.
4. Calculate the test statistic:
Single Samples t Example
4. Determine the critical value of t. Alpha =.05, two tailed
test df = N(pairs)-1 =10-1=9 ; Critical value is 1.895
5. Decision rule: The obtained t value of 0.64 does not
exceeds the critical values of 2.262 for two tailed test
under p=0.05 (because our hypothesis is non-directional)
6. Conclusion. We retain the null hypothesis that states that
there is no significant difference between this year’s
kindergarten class and the “typical” kindergarten students
in the district.

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Table A section from the Table of Critical
Value for df=9
Level of significance (p value) for one-tailed test
p values 0.10 0.05 0.025 0.01 0.005
Level of significance (p value) for two-tailed test
p values 0.20 0.10 0.05 0.02 0.01
df=9 1.383 1.833 2.262 2.821 3.2502.262
0.05
df=9
Interpretation
The mean IQ score of this year’s
kindergarten students (mean= 112.20) is
actually slightly higher than the mean
score of the district (mean=110).
The research hypothesis that was
stated in a null form (i.e., predicting
no difference between two means) is
confirmed.

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Using SPSS for dependent t-test
• Open SPSS
• Open file “SPSS Examples” (same as before)
• Go to:
– “Analyze” then “Compare Means”
– Choose “Paired samples t-test”
– Choose the two IV conditions you are
comparing. Put in “paired variables box.”
Single Sample- SPSS output
One-Sample Statistics
N Mean Std. Deviation Std. Error Mean
IQ Scores 10 112.20 10.942 3.460
One-Sample Test
Test Value = 0
95% Confidence Interval
of the Difference
t df
Sig. (2-
tailed)
Mean
Difference Lower Upper
IQ Scores 32.425 9 .000 112.200 104.37 120.03

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Task of t-test for Single samples
A report published by the Antaretic Academy of
Neurolinguistics indicates the left-handed people may be
better language learners than right-handed people. To test this
hypothesis, you did the following experiments:
In an ESL center, 250 students were randomly selected
and then assigned to two groups (left-handed vs. right-
handed). After equal amounts of instruction, you administered
a battery of language tests to all students. The information you
obtained was:
If the mean on the test for the population of ESL students is 50
a. Test whether left-handed people are better than the
population
b. Test whether right-handed people are better than the
population.
Review 6 Steps for Significance
Testing
1. Set alpha (p level).
2. State the Research
question
3. Set up hypotheses,
Null and Alternative.
4. Calculate the test
statistic (sample
value).
4. Find the critical
value of the statistic.
5. State the decision
rule.
6. State the conclusion.

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ReferencesMain Sources
Coolidge, F. L.2000. Statistics: A gentle introduction. London: Sage.
Kranzler, G & Moursund, J .1999. Statistics for the terrified. (2nd ed.). Upper Saddle
River, NJ: Prentice Hall.
Butler Christopher.1985. Statistics in Linguistics. Oxford: Basil Blackwell.
Hatch Evelyn & Hossein Farhady.1982. Research design and Statistics for Applied Linguistics.
Massachusetts: Newbury House Publishers, Inc.
Ravid Ruth.2011. Practical Statistics for Educators, fourth Ed. New York: Rowman &
Littlefield Publisher, Inc.
Quirk Thomas. 2012. Excel 2010 for Educational and Psychological Statistics: A Guide
to Solving Practical Problem. New York: Springer.
Other relevant sources
Agresi A, & B. Finlay.1986. Statistical methods for the social sciences. San Francisco,
CA: Dellen Publishing Company.
Bachman, L.F. 2004. Statistical Analysis for Language Assessment. New York: Cambridge University
Press.
Field, A. (2005). Discovering statistics using SPSS (2nd ed.). London: Sage.
Moore, D. S. (2000). The basic practice of statistics (2nd ed.). New York: W. H.
Freeman and Company.
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